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LIBRARY 
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» W*m<y «wv Nw York 3, MY. 

A^^ ^^ J^^ NEW YORK UNIVERSITY 

CO 

^ I ^1 TY ^ Institute of Mathematical Sciences 






Wl 



^ — ^ .N Division of Electromagnetic Research 



RESEARCH REPORT No. EM-88 



Turbulent Mixing Theory Applied to Radio Scattering 



RICHARD A. SILVERMAN 



CONTRACT NO. AF 19(122).42 
JANUARY, 1 956 



l^ 



NEW YORK UNIVERSITY 

Institute of Mathematical Sciences 
Division of Electromagnetic Research 

Research Report No. EM-88 



TURBULENT MIXING THEORY APPLIED TO RADIO SCATTERING 

Richard A. Silverman 



Richard A. Silverman 



/ ip'^%'^^.^ /yCc^iy\jz^ 



Morris Kline 
Project Director 



The research reported in this document has been made possible 
through support and sponsorship extended by the Air Force 
Cambridge Research Center, under Contract No. AF 19(122)-42. 
It is published for technical information only and does not neces- 
sarily represent recommendations or conclusions of the sponsor- 
ing agency. 



New York, 1956 






Abstract 



Obttkhoff s statistical theory of turbulent mixing is proposed as a replace- 
ment for the heuristic theories of Gallet and Villars-Weisskopf, and is applied 
to the problem of the scattering of radio waves by refractive index fluctuations. 
In the case of ionospheric scattering, order-of-^agnitude agreement vxith the 
observed scattered power is obtained if the refractive index fluctuations are 
attributed to electron density fluctuations produced by turbulent mixing in the 
lower edge of the E-layer, In the case of tropospheric scattering, it appears 
that order-of-raagnitude agreement with the observed scattered power can be 
obtained, except during the summer months, by attributing the refractive index 
fluctuations to temperature fluctuations. During the summer months and at low 
scattering heights, humidity and its fluctuations are expected to play a prominent 
role. Experimental and theoretical evidence is cited in favor of perennial 
fractional-degree temperature flue '-.u at ions in the troposphere. Comparison of the 
Obukhoff, Villars-Weisskopf, and Booker-Gordon models is given. 



Table of Contents 

Page 

1. Introduction 1 

2. Heuristic mixing theories 1 

3. Obukhoff 's statistical mixing theory 3 
U. Ionospheric scattering 7 

5, Tropospheric scattering. Observed scattering cross sections 9 

6. Micrometeorological interpretation 12 

7. Wavelength dependence of the scattered signal 16 

8, Conclusions 17 
Appeadix I 18 
Appendix TL 19 
References 20 



- 1 - 



1, Introduction 

Recently, both Gallet'- ^ and Villars and Weisskopf ■- -' have identified 
turbulent mixing of refractive index inhomogeneities or gradients as the fluctua- 
tion mechanism primarily responsible for the scattering of radio waves by the 
troposphere and ionosphere. In each of these papers, a heuristic theory of 
turbulent mixing is proposed. However, there is also available a statistical 
theory of turbulent mixD.ng developed by Obukhoff in 19h9» This paper is con- 
cerned with the application of Obukhoff 's theory to the radio scattering problem, 

A brief discussion of the application of Obukhoff 's theory to the radio 
scattering problem can be found in a report by Batchelor'^-', which came to the 
author's attention after he had completed the work reported here. The reader's 
attention is also drawn to the work of Krasilnikoff '- -^j who, as far back as 19U9, 
had successfully applied Obukhoff 's theory to the problem of phase fluctuations 
and angle-of-arrival fluctuations associated with the twinkling of stars, i,e,, 
to the problem of line-of-sight propagation through a randomly inhomogeneous 
atmosphere, 

2, Heuristic mixing theories 

Gallet*- -' investigates the action of turbulent mixing in the electron 

density gradient at the lower edge of the E-layer of the ionosphere. Examining 



the experimental data and calculations of Bailey et al. '-^J , he observes that 



— 

the r.m.s, fluctuation in electron density |AN/n| must be ^-ID" , in order to 



account for the observed scattered power, if the scale of turbulence L is — ^IDO 
The calculations in this reference are based on the correlation function 



exp(-r/L ), as used by Booker and Gordon*- -' , 



,M 



- 2 - 



meters . He then makes the plausible assraiption that fluctuations in electron 
density at a point are due primarily to the mixing action of the large eddies 
of size L . 30 that 

0' 

^o 



i AnA! grad N , 

where N is the average electron density and grad N its gradient. For the 



values of N and grad N at the lower edge of the E-layer, the value of | AnAJ 
calculated by tliis fonmila is indeed -^ID , 

The Villars-Weisskopf theory of turbvilent mixing applies something 
like the same heuristics to the case of small eddies, which, presiimably, 
deteiraine the difference in electron density at nearby points. To get 
the r.m.s. difference in electron density at two points a distance L apart, 

Villars and Vfeisskopf multiply the average electron density gradient by L, 

2 

This leads to an L --dependence for the mean-square difference in electron 

2/3 

density, which must be regarded as strange in view of the L ' -dependence of 

the mean-square velocity difference, Hius, it seems desirable to hare 
recourse to a statistical theory of turbiilent mixing, patterned after the 
Kolmogoroff theory, which has been so successful in explaining the micro- 
structure of the turbulent velocity field. As shown in Appendix I, statis- 
tical mixing theory and the Villars-Weisskopf theory lead to quite different 
results* 

In this paper, the symbol ^^-y means equal to within an order of magnitude, 
admittedly a somewhat loose relation, that must be handled with care. 



In an analogous way, Gallet estimates the value of | A N/N | produced by 
turbulent mixing in t he non-ad iabatic 
the E-layer and finds | An/NJ ~10"^. 



turbulent mixing in t he non-ad iabatic temperature gradient in the middle of 

-2 



- 3 - 



3, Obukhoff s statistical mixing theory 

The required statistical theorj'' of turbulent mixing already exists, 
narnely, the theory developed by Obulchoff i- ■' and elaborated by Yaglora^- -' . 
This is not the place to review the conceptual basis of Obukhoff s theory. It 
is sufficient to note that it is based on just the same sort of similarity and 
dimensionality arguments as those used by Kolmogoroff in his analysis of the 
turbulent velocity field'--', Obukhoff 's result is most simply expressed in 
terms of the structure function 



(1) D^(r) - [f(P) - f(P')]^ , 



where f(P) and f(P') axe the values at the points P and P' of the scalar quantity 
f being mixad by the turbulence (f could be electron density, temperature, etc.), 
and the overbar denotes time averaging. As a resiilt of the usual Kolmogoroff 
assuraptions of local homogeneity and local isotropy, D^(r) depends on the 
magnitude of the vector joining P and P' , but not on its direction or initial 
point, provided r is appreciably smaller than the linear dlriension of the 
largest eddy. It then follows by similarity and dimensionality considerations 
that the structure function has the fonn'- -^ 



(2) D^(r) = B^ r^/^ , 



provided that r is larger than the characteristic dimension of the smallest 
eddy, and smaller than the characteristic dimension L^ of the largest eddy, 



- u - 

Tiol* 

i.e., r must correspond to eddies in the inertial range '- ■* . 

Thus, the form of the structure function for a scalar field quantity 
mixed by turbulence is precisely that foiuid by Kolmogoroff for the vector 
velocity field itself. The existence of a large range of r for which (2) is 
valid requires that the Reynolds number of the turbulent flow be large, a condi- 
tion amply fulfilled in both the troposphere and the ionosphere. Just as in the 
case of the velocity field, the foim of. the structure function (2) itr.plies that 
the (three-dimensional) spectral density of the field f(r) has the foim- -' 



$^(k) - c^'^/^ . 



To calculate cross sections for the scattering of radio waves by 
refractive index fluctuations, we need an expression for the mean-square 



coefficient jfj^i in the Fourier expansion 



f(r) « #> fee^-^ 



'I 



7 / ^k 



vrtiere the expansion is in the scattering volume V, To relate jfj^l to the 

•JKJ- 

structure constant B-, we proceed as follows. We observe that 



(f(r ^ ?') . f (?!)?. -f ^ 1^ (1 - e-^-^), 



k 



where we have used the fact that 



The smallest eddy size for the scalar field f is not in general the s#ane as 
the smallest eddy size for the velocity field. See [7]. 

This derivation is patterned after a similar derivation for the velocity 
field in [lO], pp. 1?2-123. 

That this relation holds for all k m the inertial range is a consequence 



of the local homogeneity. See [ll] . 



- 5- 



2 



^I^^-M. = l^i?l ^ok. 



If we convert this sum to an integral, we obtain 

00 



(j(^,.,.,rr,)/ . -Jj- j; U*M.gl^ (1-^ 



sin kr )^ 



00 



- 2f S (k)(l - 2i2JSI)dk. 
Jo ^ kr 



We now assume that theBeynolds number is so high that almost the entire 

-5/3 
spectrum lies in the inertial range where the k '-^-law holds. This allows us 

to write oo 

Qf(r^r.)-f(^'))' - 2C^ J^ k-^/3 (1 - ^^ )dk. 

The integral is ^ P (l/3)r^ , so that finally 



(f(i^?') - f(r')/ = B^ r^/^ = 2.U1 C^^^-^ , 



i.e., the desired relation between the constants is 

C^ = B^/2.U1. 



Since 



r? 2n^ 



(3) If^r = ^ |f(K) , 

K 



the desired expression for If^^l in terms of the structure constant B„ is 
n^ TZ^ o 2n^ p2 „-n/3^ V o2 T^3 



-6 - 



where L = 2n/K, 



We now derive another expression for \f^ , this time in terms of 
parameters that characterize the large-scale properties of the turbulent flow. 
Our starting point is the expression 

_ CD 

Jo 

We have already noted that ^»(k) has a k '-^-dependence throughout the inertial 
range. As is customary , we assume that this dependence prevails over most 
of the spectrum, which is a good approximation at high Reynolds ntjmber. The 
value of C - is then deduced by assuming a cut-off at the largest eddy size 

h = 2"Ao- '^^^ 00 



f2 -J f^(k)dk - 1 C^;2/3^ 



k 
o 



or 

f^(k)-ff2k2/3k-^/3 



Using (3), we have finally 

(5) —t?.hii7 .^/3 K-^/3 = ^ ? L^/3 . 

^ ^o 

If we identify the structure function D„(r) with the ViUars-Weisskopf 

2 
quantity ^f- , then (U) is smaller by a factor of IDO than the corresponding 

formula (33) of [2], A detailed discussion of the source of error in [2] is 

given in [ll] . 



•K* 



See Heisenberg [12] for a similar calculation for the velocity field, 



- 7 - 



U, Ionospheric scattering 

The cross section per unit volvune for scattering by electron density 

[21 

fluctuations in the ionosphere is ■- -J 

2 

(6) cr- =~ IV 1^ cm-^ > 



where r is the classical radius of the electron (2,8 x 10~ -^ on,), and 
e 

|N^ I is the mean-square Fourier coefficient at wave number K of the electron 
density field • In terms of the wavelength X of the incident radiation, and 
the scattering angle ft, we have ^ = -^ " -^ sin -b . For the usual small values 
of 9, K^-^ and L^^ . 

Substituting from equation (5), where f is taken to be the electron 
density N, we have 

^e N^ ^11/3 
o 

In the experiment of Bailey et al. L-'j, L = l.li x ID^cm. andcr varies from 

about 5 X 10 "^ cm. (spring midnight) to about 5 x 3D~ cm," (summer noon) L J, 

To make a comparison with these values, we must deduce a plausible value of IT, 

since a direct determination of W is out of the question. If we make Gallet's 

assumption that IT is primarily due to the mixing action of the large eddies, 

then 

(7) 7 ^ ^ L^ (gradl)^. 



*In equation (6) it is assumed that the incident radiation is horizontally 
polarized. More generally, the right-hard side of (6) must be multiplied by 
sin p, when p is the angle between the incident electric field and the 
direction of scattering. 



- 8 - 



Using Gallet's value of L '^ IDO meters, and values of grad N from 1 

-3 -1 

to k electrons cm. meter , we find 

ty^ ^ lO"-"-^ tc ID"-^^ cm."''' , 

which is 10-100 times higher than representative experimental values. In 
view of the many sssumpticns implicit in this simple derivation , as well 
as the uncertainties in the values of L , grad N, and even in the experimental 
values of the cross-section"^, this must be considered 'ordsr-of-magnitude 
agreement ' " . 

If we use (li) to express the cross section in terms of the structure 



constant B„, we find 



^e ^2 11/3 
100 



The values of B^ needed to give agreement with experiment are 



Bj, ^-^ K to 5 electrons cm." . 



Direct determination of B„ transcends present experimental possibilities. 

11 /3 
Obukhoff 's theory gives a X -law for the frequency dependence of 

the scattered power, instead of the X -law predicted by Villars and 'Jeisskopf, 
___________ ^______^_______________________________________________^______ 

To define a cross section, Villars and Weisskopf arbitrarily choose the 
value 5 km. for the effective width of the scattering region. 

Obtaining values of cross sections that are larger than experimental values 
is certainly not as serious a drawback for a turbulent scattering theory as 
obtaining values that are smaller than experimental values. This point of 
view can be justified as follows. Calculations of scattering crosB sections 
are based on the assumption that the entire scattering volume is undergoing 
turbulent motion at high Reynolds number at all times. If, on the average, 
only a fraction of the scatterinf' volume is in turbulent motion at any given 
time, or if the Eeynolds number in parts of the scattering volume is so small 
that the smallest eddy size exceeds the scattering wavelength, then the 
calculated cross sections would be correspondingly smaller. 



- 9 - 



However, available experimental data does not peraiit us to choose between 

ri3i 

the exponents '- -^ , 

5« Tropospheric scattering . Observed scattering cross sections * 

In the troposphere, the appropriate formula for the scattering 
cross section per unit volume is '- -' 



n I -X |2 



where | c ^ ! is the mean-square Fourier coefficient at wave number K of the 
dielectric constant field e(r). Substituting from (U), we have 

li ® li ^ 

lOOX^ lOX^ 

where D (r) is the structure function of the dielectric constant field, 
and B is the corresponding structure constant. 

In the troposphere, e is a function of pi^ssure, humidity, and 
temperature. It has already been sho»>W.M that p^ssu,. fluctuations 
are an unimportant source of fluctuations in e, so that fluctuations in 6 
must be due primarily to humidity and temperature fluctuations. The 
meteorological equation for the dielectric constant is"- -' 

e - 1 = £ ["2.11 X 10-^ + a( 2£^ - 0.293) x 10"^], 

T *- 

where p is the pressure in mm. of mercury, T is the temperature in degjTees 
Kelvin, and a is the percentage ratio of the partial pressure of water vapor 



- ID - 



to that of dry air. Neglecting the teim -0,293, we find that the change 
in 6 due to a change in temperature is 



where A = 2,11 x 10~ , so that values of a of about 3 percent are inquired 
to double the effective value of Ao The change in e due to a change in a is 



T 



Thus, the relation between the structure function of the dielectric constant 
field and the structure functions of the temperature and humidity fields is 

D^(r) ^ (1 . ^f ( ^)2 D^(r) + 2300( ^ )\(r), 

where it is assumed that temperature fluctuations are independent of humidity 
fluctuations o The corresponding formula for the scattering cross section is 



1 (Ag)2r(i.96a)2 2^^3^32-j 
-OX^ T*^ L T ^ "^ J 



(8) ^ -fl ( ^ ^' '(1 * — )' S"' * 230OBf 1 L^/3 



For comparison with experiment, we have chosen the South Dartmouth, 
Mass. to Alpine, N.J, link, jointly operated by the Lincoln Laboratory and 
the Bell Telephone Co, The link has the following specifications'- -•: 

Frequency f = 367O Mcps, 

Wavelength X = 8,17 cm. 

Half -power beam angle for 28-foot paraboloidal 

antennas (both at transmitter and at i^ceiver), 

a ^0,65° or 0,01135 radians (azimuth), a '^ 0.70° 
a e 

or 0,0122 radians (elevation) 



- n - 



Scattering angle for midpoint of scattering 

volvme (with 3,0° antenna beam elevation), & ~ 2,75° 

or 0,Ch8 radians 

Height of midpoint of scattering voltme ~ $X)0 feet 

Path length = 188 miles 
Tlie observed scattered power varies from about -91 db, below free 
space (vrLnter) to about -61i db. belovr free space (summer) . (See [ill] j 
Fig, 2U)o To calculate the observed scattering cross sections two formulas 
are needed. The first relates the received scattered power P to the free 
space power P„ that would be received vTith the same antennas and at the 

IS 

same distance: 



^^> ^sc/^fs = i6<rv/D^. 



■JHC- 

The other is an approximation for the scattering volume V : 



(ID) V ~ ( i D sin a)^/sin 9 ^ D^a^/8». 



Using these formulas and the data given above, we calculate the 
observed scattering cross section and find that it varies from about 
h X 10~ ^ cm.'" (winter) to about 2 x 10~ cm," (summer). 

In another series of experiments on the South Dartmouth to Alpine • 
link, the transmitter and receiver antenna beams were simultaneously elevated 

These values are monthly medians of the hourly median values of the 
signal level, 

"^ 2 

More accurately, V r^ e a 

8 © 



- 12 - 



in 0,2 steps. In this way, the midpoint of the scattering volume was raised 
from about 5300 feet to about lUOOO feet. The results of these tests are 
shown in Table I , 



Scattering cross 
section per unit -^^rrw- 
voliwie in cin,~rxl0 ■, 



.375 to 190 
5.95 
5.1. 
3.05 

0.855 

0.30 
0.205 

o.iU 









Table 


I 


Antenna beam 
elevation in 
degrees 


Midpoint 
height in 
feet 


Midpoint 
scattering 
angle in 
radians 


Received power 
in db. below 
free space 


0.0 




5tK)0 


.QU8 


91 to 6U 


0,0 




5000 


.0U8 


79 


0.2 




6500 


.055 


80 


o.U 




8000 


.062 


83 


0.6 




95bo 


.069 


89 


0.8 




11000 


.076 


9h 


1.0 




12500 


.083 


96 


1.2 




DjOOO 


.090 


98 



6, Micrometeorological interpretation . 

The available measurements of the microstructure of the temperature 
field in the troposphere are very meagerj the author knows of no direct 
measurements of the microstructure of the humidity field. In view of this 
situation, we shall calculate the values of B„ needed to account for the observed 
scattering, neglecting the effects of humidity, by setting a and B equal to 

The first entry in Table I corresponds to the annual variation of the received 
scattered povrer, which, as already noted, varies from about -91 to about -6h db, 
below free space. The rest of the table corresponds to the beam elevation ex- 
periments, which were made when the povjer received at 0,0° elevation was -79 db, 
(The same remarks apply to Table II), 



•JHt 



The reader will realize that due to the large number of approximations made, 
it is impossible to obtain more than rough estimates of the scattering cross 
sections. Two significant figures are given in the tables only to prevent 
unnecessary round-off errors. 



- 13 - 



zero, \-Je shall then compare the values of B^ so obtained with experimental 
estimates^ if a larger value of B_, is required than seems reasonable, we shall 
see whether the deficit in scattered power can be made up by the humidity terms. 
fit large scattering heights, where the dryness of the air precludes any 
important humidity effects, it should be possible to account for the observed 
scattered power by temperature fluctuations alone. 

Setting a and B equal to zero, we solve (8) for the values of B„(Ap/r ) 
needed to give agreem.ent with the observed cross sections. Then we calculate 
appropriate values of p and T for the various scattering heights, using the 



following particularizations of formulas given by Mitra 



m. 



T = 293 - 1.521; h, 

p = 760(1 - .a'j52h)^*^^ , 



where h is the height above the earth's surface in thousands of feet. Using 
these values of p and T, we calculate corresponding values of B„, The results 
are sujimarized in Table II, 









Table II 








Antenna 
beam 

elevation 
in degrees 


Midpoint 
height 
in feet 


Tempera- 
ture in 
degrees 
K. 


Pressure 
in ram, 
mercury 


Value of I 
needed tc 
agreement 
experiment 


i^CApA^) 

) give 
with 
; X 10° 


Correspond- 
ing value 
of B^^-^ 


0.0 


^0 


28$ 


63U 


1.1 to 


2U 


.005 to .15 


0.0 


^0 


285 


63U 


i;,2 




.025 


0.2 


65bo 


283 


600 


5.1 




.030 


OoU 


8000 


281 


568 


U.8 




.030 


0,6 


95)0 


279 


537 


3.1 




.020 


0,8 


nooo 


276 


507 


2.2 




.015 


1.0 


12^0 


27li 


U79 


2.1 




.015 


1.2 


lUOOO 


272 


U52 


2.0 




.015 



^These formulas are based on the values p = 760 mm. and T = 293 K. at the earth's 
surface, and the temperature lapse rate of -5°K./l<m. 

i;-;;-To nearest multiple of .005 (or.05). 



-11,- 



Examination of Table II shows that values of the ter.persture striacture 
constant B_ of about 2 to 3 x 10 are needed to explain the observed scattered 
power at the time of the year when the antenna beam elevation experiments 
were performed. Let us examine the micrometeorological evidence in favor of 
values of this order of magnitude. 

First, there is the experiment of Krechmer^ -' , who, in a series of 

delicate m.easurements with low-inertia resistance thermometers, verified the 

2 /*? —2* 

Obukhoff r -law, and found B„ /^ 3 x ID , However, it is admittedly- 
questionable to extrapolate Krechmer' s value of B^^, which was measured near 
the earth's surface, to tropospheric scattering heights, since, near the 
earth's surface, neither the turbulent flovr nor the temperature regime can 
be regarded as typical of the troposphere as a whole. On the other hand, the 
agreement is striking. 

Secondly, we can borrow Gallet's idea of turbulent mixing in non- 
adiabatic temperature gradients. The actual temperature lapse rate in the 
troposphere is not adiabaticj the adiabatic lapse rate is — ■ -10 K,/^,, 
whereas the observed lapse rate is /^-5°K/l<n,L -^J Thus, for the purpose of 
inducing temperature fluctuations, the turbulence works in a lapse rate of 
r^ -^/km. Comparing (U) and (5), we have 

The analogue of equation (?) for the ionospheric case is 

T^ - i L^(grad T)^, 

*Obukhof?'^ estimated B„ ^^ 2 to U x 10" . Following Obukhoff, Krasilnikoff 
used the value 3 x 10" in his line-of-sight calculations'- K 



-15- 



where grad T is the non-adiabatic temperature gradient, Choosinp- for L the 

-" o 

value of 10 cm, recommended by Batchelor'- -', wj find 



lATl ^ 3/10° K, 
and 

B^ ^ 3x10"', 

in complete agreement with the other estimates of B„, This argument suggests 
that away from the earth's surface, B^ will have very much the same value over 
quite a range of heightsj according to Table II, this is indeed the case in 
the height range 11,OCO to lli,000 feet. 

The presence of fractional-degree temperature fluctuations in the 

ri7i 

troposphere and stratosphere has been amply verified by Scrase. '- -' In a 
radiosonde experiment in late June, he found that the r,m,s, temperature 
fluctuation was ^ 0,8y^F, in the height range 1-12 km., '^0,61; F. in the 
range 12-21 km,, and ^^0,29 F, in the range 21-31 km. 

To account for the greatly enhanced scattered power in the summer 
months, we must invoke humidity, for values of B_, such as .15 (see Table II) 
are about five times too large. As we have seen, humidity plays a twofold 
role. Not only do humidity fluctuations enhance the scattered signal directly, 
but temperature fluctuations cause stronger dielectric constant fluctuations in 
moist air than in dry air. By combining the effects of humidity with enhance- 
m.ent of tem.perature fluctuations themselves, it should be possible to explain 
the enhancement of the scattered signal during the summer months. Until 
further experimental infonnation has been acquired, there does not seem to 
be much more that can be said about the role of humidity fluctuations, Hoxrever, 
simultaneous measurements with low- inertia thermometers and microwave refracto- 
meters should permit a detailed analysis of the micros tructure of the humidity 



- 16 - 



field,* 



7. Wavelength dependence of the scattered signal 

An examination of (8) shows that according to the Obukhoff theory, 

the wavelength dependence of the scattering cross section per unit voliime 

-1/3 
should obey a \ -law, Tixperimental verification of this law, which differs 

only weakly from the \ -law of Booker and Gordon '■ ^ will probably not be 

possible until a scaled experiment at ^videly separated frequencies has been done, 

ri9i 

Although such an experiment is under consideration"- -• , it has yet to be 
performed . However, data already available counterindicates the \-law of 

the Villars-Weisskopf mixing theory, 

[20! 

Bullington et al. -' measured siinul taneous transmissions at $0^ mc, 

and I4D9O mc, over a 1^ nautical mile path in Newfoundland for a full year. The 
power received at UO^O mc, varied seasonally from about 12 to 20 db» below that 
received at 5t'5 mc. Both circuits used antennas of the same size (28 foot 
paraboloids) so that the beamwidth at 505 mc, should be roughly 8 times that 

at lj090 mc. Thus, by equation (ID) the scattering volume at 5o5 mc. is 

•a 

roughly 8 times that at ljD90 mc. If the scattering cross section per unit 

-I/3 
volume obeys a X ' -law, then by (9), the received power should obey a 

0/0 
X ' -law, i,e,, the power received at hP90 mc. should be about 26 db. below 

that received at 505 mc. If, on the other hand, the scattering cross section 

obeys the ViUars-Weisskopf X-law, then the received power should obey a \ -law, 

i,e,, the power received at li090 mc, should be about Ul db. below ohat received 

Of course, the refractive index field itself, as measured directly on the 
microwave refractometer should follow the r^/^-iaw, gefjactometer measure- 
ments by Birnbaum and Bussey seem to verify this law 



Refr? 
.L18] 



■5H<-In the case of ionospheric scattering, scaled experiments have been carried 
out. [13J 



- 17 - 
at 5b5 mc, 

/.ctually, these considerations are much too iinsophisticated. Before the 
exact wavelength dependence of the scattered power can be determined we must consider 
(ainong other factors) the effects of antenna gain degradation and the effects of 
scattering angle variation in the scattering volume. In any event, it is clear that 
both the Booker-Gordon X -law and the \~ -law that follows from Obukhoff 's statis- 
tical mixing theorj' are easier to reconcile with experiment than the Villars-l"Jeisskopf 
X-law, 

8. Conclasions 

There are many experiments that may help test this theory of radio scattering, e.g. 

1, Study of the seasonal variation of tropospherically scattered signals as a 

function of scattering height, 

-1/3 

2, Test of the X -law by scaled scattering experiments, 

3, Investigation of the microstructure of the temperature, humidity, and 

2/3 

refractive index fields in the troposphere j attempts first to verify the r ' -law, and 

then to measure the structure constants of the temperature and humidity fields as 

functions of height, season, time of day, etc. 

2/3 

h» If the refractive index field n obeys the r ' -law, then one should check 

whether the observed power in a scattering experiment can be calculated from values of 
the structure constant of the refractive index field measured in (or near) the scatter- 
ing volume. The theoretical expression for the scattering cross section per unit rolume 
is 

-ii-p b2l^/3 , 
IDT? ^ 

where the factor of )4 is needed to convert dielectric constant to refractive index. 

Doubtless, other relevant experiments will suggest themselves to the reader, 

■**■ Both of these effects bring the X^-law more in line with the exjDeriments of 
Bullington et al. Because of the antenna gain degradation, the X^-law is 
multiplied by a law weaker than X3 when we convert the scattering cross section 
per unit volume to received power. Because of the scattering angle variation, 
the lower frequencies scatter less than if the scattering were all at the nominal 
midpoint scattering angle. However, the discrepancy with experiment that has to be 
made up is least for the X~-'-/3-law, 



- 18 - 

Appendix I « Comparison of the Obukhoff and Villars-Weisskopf models. 

The Villars-V/eisskopf heuristic theory and the Obtikhoff statistical theory of 
turbulent mixing lead to different expressions for the structure function. In this 
appendix we compare the two expressions. 

According to the Villars-Weisskopf theory 

D^/2(L) - !f(P) - f(P»)| -^ (grad f) L, 

where L is the distance between the points P and P'. On the other hand, according 
to the Obukhoff theory 



where 



b2 . 5 7/ l//3 ^ (grad f )2 L^^3 ^ 



2 

if we attribute f primarily to the mixing action of the large eddies of size L . 

Thus, the Obukhoff expression for the structure function is 
D^^d) '^ (grad f) l;/3 L^/5 , 

which agrees with the Villars-Weisskopf expression only for L -^ L . For smaller 

1/2 
values of L, the Ob\ikhoff theory gives larger values of D„' (L) than the Villars- 
Weisskopf theory, i.e., the action of the turbulence produces local gradients 
steeper than the prevailing average gradient by a factor (l-A) • Thus, the 
Obukhoff model can be expected to produce more radio scattering from a given 
refractive index gradient than the Villars-Weisskopf model. 

Finally, it should be noted that all mean-square quantities that appear in this 
paper are variances, i.e., the random variables f(P), f(P')> etc., can be regarded 
as centered about their means. This is Justified by the dynamically passive nature 
of the quantity f, which assures that the turbulent regime is negligibly influenced 
even by considerable changes in 1 over distances of order L . 



- 19 - 

Appendix II . Comparison of the Obukhoff and Booker-Gordon models. 

By the Booker-Gordon model w« mean the use of the exponential correlation 
function exp(-r/L ). For this correlation function, tlrie scattering cross section 
per unit volume is 

where L ^ X/O and n is the variance of the refractive index. On the other hand, 
combining the general fornmla for the scattering cixiss section per unit volume 

2 



cr- 



^ ''^''' 



with equation (5), we find that according to thie Obukhoff theory 



7, l1^/3 
o 



Ihe ratio of the Obukhoff and Booker-Gordon expressions is 



f^BG 



o' 



Thus, the Obukhoff model predicts more scattering from a given variance of 

1/3 
refractive index than the Booker-Gordon model by a factor (L A) • (Compare 

with analogous result in Appendix I.) This fact is striking in view of recent 

l2ll 
failures to account for the observed scattering using the Booker-Gordon model.*- -■ 



- 20 - 



References 



[ij Gallet, R.M, - Aerodynamical mechanisms producing electron density 

fluctuations in turbulent ionized layers; Proc, I.R.E,, 
U3, 12U0 (19^5). 

[2] Villars, F, and Weisskopf, V,F, - On the scattering of radio waves by turbulent 

fluctuations of the atmosphere; Proc, I.R.E., U3 j 
1232 (1955). 

[3] Batchelor, G.K, - The scattering of radio waves in the atmosphere by 

turbulent fluctuations in refractive index; Technical 
Report No, 26, School of Electrical Engineering, Cornell 
University, 1$ September, 1955« 

[U] Krasilnikoff, V.A, - On fluctuations of the anglR of arrival in the phenomenon 

of twinkling of stars; Doklady Akad. Nauk SSSR, 6|, 291, 
(19U9). 

- On the influence of pulsations of refractive index in 
the atmosphere on the propagation of ultrashort radio 
waves; Izv. Akad. Nauk SSSR, Ser. Geograf, i Geofiz., 
13, 33 (19U9). 

[S\ Bailey, D.K,, Bateman, R., Berkner, L.V., Booker, H.G., Montgomery, G.F,, 

Purcell, E.M. , Salisbury, W.W. , and Wiesner, J.B. - 
A new kind of radio propagation at very high frequencies 
observable over long distances; Phys. Rev., 86 , Ihl, 
(1952). 

[6] Booker, H.G, and Gordon, W,E. - A theory of radio scattering in the troposphere; 

Proc. I.R.E., 38, 1^1 (1950), 

[7] Obukhoff, A.M. - The structure of the temperature field in turbulent 

flow; Izv. Akad. Nauk SSSR, Ser. Geograf. i Geofiz., 
13, 58 (19U9). 

[§] Yaglora, A.M. - On the local structure of the temperature field in 

turbulent flow; Doklady Akad. Nauk SSSR, 69, 7U3 (19U9). 



- 21 - 



[9j Kolmogoroff, A.N. 



- The local structure of turbulence in incompressible 
viscous fluid for very large Reynolds numbers 3 Doklady 
Akad. Nauk SSSR, 30, 301 (I9UI). 

- Dissipation of energy in the locally isotropic turbulence, 
ibid., 32, 16 (I9hl). 



[lOj Batchelor, G.K, 



- The tlieory of homogeneous turbulence^ Cambridge 
University Press, (1953). 

[11] Balser, M, and Silverman, R.A, - On the scattering of radio waves by 

refractive index fluctuations. Part I, Phenomenological 
theory, in preparation, 

[12J Heisenberg, W, - Zur statistischen Theorie der Turbulenzj Zeits. f. 

Physik, 12U, 628, (I9li8). 

[l3_l Bailey, D.K,, Bateman, R., and Kirby, R.C. - Radio transmission at VHF by 

scattering and other processes in the lower ionosphei^^ 
Proc. I.R.E., U3, 1181, (1955). 

[lli] Chisholm, J.H., Portmann, P. A., deBettencourt, J.T., and Roche, J.F. - 

Investigations of angular scattering and multipath properties 
of tropospheric propagation of shoii; radio waves beyond the 
horizon; Proc. I.R.E., h3, 1317, (1955). 

- The upper atraospherej The Asiatic Society, Calcutta, 
2nd edition, (1952). 



[15] Mitra, S.K. 
[16] Krechmer, S.I. 



- Investigation of micropulsations of the temperature 
field in the atmosphere; Doklady Akad. Nauk SSSR, _8U, 
^^, (1952). 



[17] Scrase, F. J. 



- Turbulence in the upper air, as shown by radar-wind and 
radiosonde measurements; Quarterly Journal of the Royal 
Meteorological Society, 80, 369, (195U). 

[18] Bimbaum, G. and Bussey, H.E, - Amplitude, scale, and spectrum of refractive 

index inhomogeneities in the first 125 meters of the 
atmosphere; Proc. I.R.E,, U3, 1U12 (1955). 

[19] Gordon, W.E. - Private comrauni cation. 



/ 

- 22 - 

[20] Bullington, K., Inkster, W.J., and Durkee, k,L, - Results of propagation 

tests at 505 mc and 14390 mc on beyond-horizon pathsj 
Proc. I.R.E., U3, 1306 (1955). 

[21] Grain, C.M,, private camminication. 





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Turbulent mixing 
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Turbulent mixing theoi?y 
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